Second, and perhaps most importantly, we propose that the (generalised) thermodynamics of the soliton gas can be described in terms of the classical Thermodynamic Bethe Ansatz (TBA). That is, we propose that the general form expressed in [1]} for the thermodynamics of integrable many-body systems holds for soliton gases as well. In this, a piece of information about the gas is required, that was not discussed yet in the literature on soliton gases: the “statistics" of the solitons. We propose that solitons are associated to the Maxwell-Boltzmann statistics, like classical indistinguishable particles. As a result, we identify the key thermodynamic quantities associated with the KdV soliton gas (including the entropy, the free energy, the temperature). Using the general results of GHD, we then infer the static covariances, the space-integrated two-point correlation functions involving densities and currents. We confirm our conjecture of the Maxwell-Boltzmann statistics by comparing static covariances against numerical calculations.
